Density estimation

Keywords, protein folding, tertiary structure, potential energy surface, global optimization, empirical potential, residue potential, surface potential, parameter estimation, density estimation, cluster analysis, quadratic programming... 

The only density estimators discussed in the protein literature are histogram estimates. However, these are nonsmooth and thus not suitable for global optimization techniques that combine local and global search. Moreover, histogram estimates have, even for an optimally chosen bin size, the extremely poor accuracy of only, for a sample of size n. The theo-... 

We therefore use smooth density estimation techniques that are more reliable than the histogram estimates. To improve the reliability for rare amino acid pairs, we use clustering techniques that identify similar pairs that can be modeled by the same density. 

The polynomials Wa q) q = qik) and Wy q) (q = qi) needed to specify the pair and surface potentials are constructed from the set of such q realized in a data base of 266 proteins with a total of 46100 residues by means of density estimation techniques. 

One therefore needs a smooth density estimation techniques that is more reliable than the histogram estimates. The automatic estimation poses additional problems in that the traditional statistical techniques for estimating densities usually require the interactive selection of some smoothing parameter (such as the bin size). Some publicly available density estimators are available, but these tended to oversmooth the densities. So we tried a number of ideas based on numerical differentiation of the empirical cdf to devise a better density estimator. 

P. Hall and J.S. Marron, Lower bounds for bandwidth selection in density estimation, Probab. Th. Rel. Fields 90 (1991), 149-173. 

M. C. Jones, J.S. Marron and S.J. Sheather, Progress in datarbased bandwidth selection for kernel density estimation, Comput. Statist. 11 (1996), 337-381. 

Figure 1. The global carbon cycle. Estimates of reservoir size and annual fluxes are from Post et al. (4), Vegetation carbon reservoir was estimated from latest carbon density estimates. All values except the atmospheric reservoir are approximate only. All values are in gigatons. Fluxes are next to the arrows and are in gigatons ear.

Silverman, B. W., Density Estimation for Statistics and Data Analysis. Chapman Hall, New York, 1986. 

Fig. 33.14. Density estimate for a test set using normal potential functions (univariate case).

D. Coomans and D.L. Massart, Alternative K-nearest neighbour rules in supervised pattern recognition. Part 2. Probabilistic classification on the basis of the kNN method modified for direct density estimation. Anal. Chim. Acta, 138 (1982) 153-165. 

H. van der Voet and D.A. Doornbos, The improvement of SIMCA classification by using kernel density estimation. Part 1. Anal. Chim. Acta, 161 (1984), 115-123 Part 2. Anal. Chim. Acta, 161 (1984) 125-134. 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

Step 13. Generate the output table of temperature, observed densities, estimated uncertainties, and difference between observed and calculated densities and arrange it in order of year of publication with authors. For data from a particular source, arrange in order of temperature. 

The density estimates in Table 7.1 show a distinction between the structures of the planets, with Mercury, Venus, Earth and Mars all having mean densities consistent with a rocky internal structure. The Earth-like nature of their composition, orbital periods and distance from the Sun enable these to be classified as the terrestrial planets. Jupiter, Saturn and Uranus have very low densities and are simple gas giants, perhaps with a very small rocky core. Neptune and Pluto clearly contain more dense materials, perhaps a mixture of gas, rock and ice. 

Tune, L. E., Wong, D. F., Pearlson, G. et al. Dopamine D2 receptor density estimates in schizophrenia a positron emission tomography study with C-N-methylspiperone. Psychiatr. Res. 49 219-237,1993. 

In the absence of an assumed underlying normal distribution, simple bivariate plotting does not lead to an estimate of the true extent of the parent isotope field. This is particularly a problem if only relatively few samples are available, as is usually the case. Kernel density estimation (KDE Baxter et al., 1997) offers the prospect of building up an estimate of the true shape and size of an isotope field whilst making few extra assumptions about the data. Scaife et al. (1999) showed that lead isotope data can be fully described using KDE without resort to confidence ellipses which assume normality, and which are much less susceptible to the influence of outliers. The results of this approach are discussed in Section 9.6, after the conventional approach to interpreting lead isotope data in the eastern Mediterranean has been discussed. 

Figure 9.12 Kernel density estimate of the lead isotope data for part of the Troodos ore field, Cyprus (data from Gale et al., 1997). The superimposed Oxford ellipse has been used to represent the Cyprus ore field in several publications. (From Scaife et al., 1999 Figure 6, with permission from the first author.)...

Baxter, M.J., Beardah, C.C. and Wright, R.V.S. (1997). Some archaeological applications of kernel density estimates. Journal of Archaeological Science 24 347-354. 

One problem encountered in solving Eq. (11.12) is the modeling of the prior distribution P x. It is assumed that this distribution is not known in advance and must be calculated from historical data. Several methods for estimating the density function of a set of variables are presented in the literature. Among these methods are histograms, orthogonal estimators, kernel estimators, and elliptical basis function (EBF) estimators (see Silverman, 1986 Scott, 1992 Johnston and Kramer, 1994 Chen et al., 1996). A wavelet-based density estimation technique has been developed by Safavi et al. (1997) as an alternative and superior method to other common density estimation techniques. Johnston and Kramer (1998) have proposed the recursive state... 

Following Johnston and Kramer (1998) and using an EBF probability density estimator, the form of the estimator for E x is... 

Johnston, L. P. M., and Kramer, M. A. (1994). Probability density estimation using elliptical basis function. AIChE J. to, 1639-1649. 

Safavi, A., Chen, J., and Romagnoli, J. A. (1997). Wavelet-based density estimation and application to process monitoring. AIChE J. 43, 1227-1241. 

Scott, D. W. (1992). Multivariate Density Estimation Theory Practice and Visualisation. Wiley, New... 

This represents an upper limit for the dimensions of the nucleus. Compared with the estimates for the size of the atom, obtained from kinetic theory calculations on gases, which are typically 4x10 9 m. we can see that the nucleus is very small indeed compared to the atom as a whole - a radius ratio of 10-5, or a volume ratio of 10 15, which supports Rutherford s observation that most of an atom consists of empty space. We can also conclude that the density of the nucleus must be extremely high - 1015 times that encountered in ordinary matter, consistent with density estimates in astronomical objects called pulsars or neutron stars. 

Kernel density estimate of the lead isotope data for part... 

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